Transcript File - Mr. Flynn`s Math Website

CHAPTER 1
Problem Solving and
Critical Thinking
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1.1
Inductive and Deductive Reasoning
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2
Objectives
1. Understand and use inductive
reasoning.
2. Understand and use deductive
reasoning.
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Inductive Reasoning
• The process of arriving at a general conclusion
based on observations of specific examples.
• Definitions:
– Conjecture/hypothesis: The conclusion formed as
a result of inductive reasoning which may or may
not be true.
– Counterexample: A case for which the conjecture
is not true which proves the conjecture is false.
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Strong Inductive Argument
• In a random sample of 380,000 freshman at 772 fouryear colleges, 25% said they frequently came to class
without completing readings or assignments. We can
conclude that there is a 95% probability that between
24.84% and 25.25% of all college freshmen
frequently come to class unprepared.
• This technique is called random sampling, discussed
in Chapter 12. Each member of the group has an
equal chance of being chosen. We can make
predictions based on a random sample of the entire
population.
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Weak Inductive Argument
• Men have difficulty expressing their feelings.
Neither my dad nor my boyfriend ever cried in
front of me.
– This conclusion is based on just two observations.
– This sample is neither random nor large enough to
represent all men.
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Example 2a: Using Inductive Reasoning
• What number comes next?
• Solution: Since the numbers are increasing
relatively slowly, try addition.
– The common difference between each pair of
numbers is 9.
– Therefore, the next number is 39 + 9 = 48.
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Example 2b: Using Inductive Reasoning
• What number comes next?
• Solution: Since the numbers are increasing
relatively quickly, try multiplication.
– The common ratio between each pair of numbers
is 4.
– Thus, the next number is: 4  768 = 3072.
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Inductive Reasoning: More than one
Solution!
2, 4, ?
What is the next number
in this sequence?
Is this illusion a wine
Goblet or two faces
looking at each
other?
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– If the pattern is to add
2 to the previous
number it is 6.
– If the pattern is to
multiply the previous
number by 2 then the
answer is 8.
• We need to know one
more number to
9
decide.
Example 3: Fibonacci Sequence
• What comes next in this list of numbers?
1, 1, 2, 3, 5, 8, 13, 21, ?
• Solution: This pattern is formed by adding the
previous 2 numbers to get the next number:
• So the next number in the sequence is:
13 + 21 = 34
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Example 4: Finding the Next Figure in a
Visual Sequence
• Describe two patterns in this sequence of
figures. Use the pattern to draw the next
figure.
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Example 4 continued
• Solution: The first pattern concerns the
shapes.
– We can predict that the next shape will be a Circle
• The second pattern concerns the dots within
the shapes.
– We can predict that the dots will follow the pattern
from 0 to 3 dots in a section with them rotating
counterclockwise so that the figure is as bel
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Deductive Reasoning
• The process of proving a specific conclusion
from one or more general statements.
• Theorem: A conclusion proved true by
deductive reasoning
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An Example in Everyday Life
Everyday Situation
Deductive Reasoning
One player to another
in Scrabble. “You
have to remove those
five letters. You can’t
use TEXAS as a
word.”
General Statement:
All proper names are prohibited
in Scrabble.
TEXAS is a proper name.
Conclusion:
Therefore TEXAS is prohibited
in Scrabble.
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Example 5: Using Inductive and Deductive
Reasoning
Using Inductive Reasoning, apply the rules to specific
numbers. Do you see a pattern?
Select a number
4
7
11
Multiply the number
by 6
4 x 6 = 24
7 x 6 = 42
11 x 6 = 66
Add 8 to the product
24 + 8 = 32
42 + 8 = 50
66 + 8 = 74
Divide this sum by 2
32
 16
2
Subtract 4 from the
quotient
16 – 4 = 12
50
 25
2
25 – 4 = 21
74
 37
2
37 – 4 = 33
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Example 5 continued
• Solution:
– Using Deductive reasoning, use n to represent the
number
Does this agree with your inductive hypothesis?
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